Some results of semilocally simply connected property 1. Introduction
... At first, by use of Lemma (7.6.15) and Lemma (7.6.13) of [7] we state a simple proof for the following Theorem. Theorem 2.1 Let X be a topological space and f : S n−1 → X be a continuous map. If X ′ = X ∪f E n and Y be a CW-complex with dimension little than n, then for every continuous map g : Y → ...
... At first, by use of Lemma (7.6.15) and Lemma (7.6.13) of [7] we state a simple proof for the following Theorem. Theorem 2.1 Let X be a topological space and f : S n−1 → X be a continuous map. If X ′ = X ∪f E n and Y be a CW-complex with dimension little than n, then for every continuous map g : Y → ...
Contents - POSTECH Math
... The Zn is a normal sub group of Rn , so that Tn ≡ Rn /Zn ≃ S1 × · · · × S1 is the n-torus. Definition 1.5 Let M1 and M2 be 2-manifolds. Let Di ⊆ Mi be imbedded disks, and Mi′ = Mi −int(Di ), i = 1, 2. Then, ∂Mi′ in Mi are homeomorphic to ∂Di as they are circles, so that they can be glued together by ...
... The Zn is a normal sub group of Rn , so that Tn ≡ Rn /Zn ≃ S1 × · · · × S1 is the n-torus. Definition 1.5 Let M1 and M2 be 2-manifolds. Let Di ⊆ Mi be imbedded disks, and Mi′ = Mi −int(Di ), i = 1, 2. Then, ∂Mi′ in Mi are homeomorphic to ∂Di as they are circles, so that they can be glued together by ...
Math 440, Spring 2012, Solution to HW 1 (1) Page 83, 1. Let X be a
... X such that for each open set U of X and each x in U , there is an element C of C such that x ∈ C ⊂ U. Then C is a basis for the topology of X. Note that the standard topology on R is generated by open intervals (a, b) where a, b ∈ R. So by definition of a topology generated by a basis, for any open ...
... X such that for each open set U of X and each x in U , there is an element C of C such that x ∈ C ⊂ U. Then C is a basis for the topology of X. Note that the standard topology on R is generated by open intervals (a, b) where a, b ∈ R. So by definition of a topology generated by a basis, for any open ...
Continuous Functions in Ideal Bitopological Spaces
... Definition 3.1: A function f: (X, τ1, τ2, I) (Y, σ1, σ2) is said to be (i,j)-I - continuous if f-1(V) is (i,j)-I - open in X for every σi -open set V in Y; i, j=1, 2, i ≠ j Definition 3.2: A function f: (X, τ1, τ2, I) (Y, σ1, σ2) is said to be (i,j)- precontinuous if f-1(V) is (i,j)- preopen in ...
... Definition 3.1: A function f: (X, τ1, τ2, I) (Y, σ1, σ2) is said to be (i,j)-I - continuous if f-1(V) is (i,j)-I - open in X for every σi -open set V in Y; i, j=1, 2, i ≠ j Definition 3.2: A function f: (X, τ1, τ2, I) (Y, σ1, σ2) is said to be (i,j)- precontinuous if f-1(V) is (i,j)- preopen in ...
Chapter 4 Note Packet
... In the scoring system of some track meets, for first place you get 5 points, for second place you get 3 points, for third place you get 2 points, and for fourth place you get 1 point. This scoring system can be shown as ordered pair, {(1,5), (2,4), (3,2), (4,1)} You can also show relations in other ...
... In the scoring system of some track meets, for first place you get 5 points, for second place you get 3 points, for third place you get 2 points, and for fourth place you get 1 point. This scoring system can be shown as ordered pair, {(1,5), (2,4), (3,2), (4,1)} You can also show relations in other ...
Lecture8.pdf
... Any one-to-one function has an inverse. A function is one-to-one if no two ordered pairs in the function have the same second component. In other words, one-to-one functions do not have any repeated values. Consequently, y sin x is not one-to-one. A quick test for one-toone correspondence is the h ...
... Any one-to-one function has an inverse. A function is one-to-one if no two ordered pairs in the function have the same second component. In other words, one-to-one functions do not have any repeated values. Consequently, y sin x is not one-to-one. A quick test for one-toone correspondence is the h ...